arxiv: v1 [math.pr] 9 May 2008

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1 Degree-distribution stability of scale-free networs Zhenting Hou, Xiangxing Kong, Dinghua Shi,2, and Guanrong Chen 3 School of Matheatics, Central South University, Changsha 40083, China 2 Departent of Matheatics, Shanghai University, Shanghai , China 3 Departent of Electronic Engineering, City University of Hong Kong, Hong Kong, China Dated: May 0, 2008 arxiv: v [ath.pr] 9 May 2008 Based on the concept and techniques of first-passage probability in Marov chain theory, this letter provides a rigorous proof for the existence of the steady-state degree distribution of the scalefree networ generated by the Barabási-Albert BA odel, and atheatically re-derives the exact analytic forulas of the distribution. The approach developed here is quite general, applicable to any other scale-free types of coplex networs. PACS nubers: Hc, Ln, Ge, Da Introduction. The intensive study of coplex networs is pervading all inds of sciences today, ranging fro physical to biological, even to social sciences. Its ipact on odern engineering and technology is proinent and will be far-reaching. Typical coplex networs include the Internet, the World Wide Web, wired and wireless counication networs, power grids, biological neural networs, social relationship networs, scientific cooperation and citation networs, and so on. Research on fundaental properties and dynaical features of such coplex networs has becoe overwheling. In the investigation of various coplex networs, the degree distributions are always the ain concerns because they characterize the fundaental topological properties of the underlying networs. Noticeably, for a ring-shape regular graph [] of whatever size, where every vertex is connected to its K nearest-neighboring vertices, all vertices have the sae degree K. For the well-nown Erdös-Rényi rando graph odel [2] with n vertices and edges, the degree distribution of vertices is approxiately Poisson with ean value 2/n. For the sall-world networ proposed by Watts and Strogatz [], the degree distribution of vertices also follows Poisson distribution approxiately. A coon feature of the above odels is that the degree distribution of vertices has a characteristic size. In contrast, Barabási and Albert [3] found that for any real-world coplex networs, e.g., the WWW, the fraction P of vertices with degree is proportional over a large range to a scale-free power-law tail: γ, where γ is a constant independent of the size of the networ. Thus, the fraction P of vertices with degree is referred to as the degree distribution of a scale-free networ. To explain this phenoenon, they proposed the following networ-generating echanis [3], nown as the BA odel: starting with a sall nuber 0 of vertices, at every tie step we add a new vertex with 0 edges that lin the new vertex to different vertices already present in the syste. To incorporate preferential attachent, we assue that the probability Π that a new vertex will be connected to a vertex depends on the connectivity i of that vertex, so that Π i i / j j. After t steps the odel leads to a rando networ with t + 0 vertices and t edges. In [3], coputer siulation showed that for the BA odel the degree distribution of the networ has a power law for with the exponent γ 2.9 ± 0.. In [4], a heuristic arguent based on the ean-field theory led to an analytic solution P 2 2 3, naely γ 3. To derive the following dynaic equation: i t Π i i 2t, ii, it was assued [4] that the probability can be interpreted as a continuous rate of change of i for an existing vertex with degree i to receive a new connection fro the new vertex is exactly equal to Π i, which is siultaneously proportional to both the degree i t of the existing vertex i and the nuber of the new edges that the new vertex brings in, at tie t. For notational convenience, this assuption will be siply referred to as the Π-hypothesis in this paper. In all the consequent wors related to the BA odel, this Π-hypothesis plays a fundaental role. For exaple, Krapivsy et al. [5] replaced the degree i t of vertex i at tie t by the total nuber N t of degree- vertices over the whole networ at tie t, thereby obtaining its rate equation dn t dt N t N t N t + δ, where δ accounts for new vertices bringing in new edges. In this study, the Π-hypothesis was adopted in the derivations. Assuing that the steady-state degree distribution exists, using the law of large nubers N t t P as t, they showed that the difference equation of P has an analytic solution P for the BA odel with. They also pointed out that only the linear preferential attachent schee can lead to the scale-free structure but any nonlinear one will not. Dorogovtsev et al. [6] considered i t as a rando variable and defined P, i, t to be the probability that vertex i has exactly edges at tie t, where vertex i is the vertex that was being added to the networ at tie t i, i, 2,. Moreover, they used the average of all vertex degrees as the networ degree: P, t t t i P, i, t. They introduced a ore general attraction odel and allowed ultiple edges between vertices,

2 2 where each new vertex has an initial attraction degree A. Siultaneously, new directed edges coing out fro non-specified vertices are introduced with the probability Π, therefore A + q with q being the in-degree of vertices. Consequently, when every new vertex is the source of the new edges lie in the BA odel, the attraction odel aes ore sense than the BA odel. They first arrived at the aster equation of P, i, t and then by suing all i s together they were able to derive the following equation: P, t + 2t +δ + O P, t + P, t. t 2t P, t To that end, by assuing the existence of P [note that actually an additional assuption of li t[p, t + P, t] 0 is also needed], they obtained a difference equation for P. Finally, solving the equation gave an analytic solution P Here, it should be pointed out that if ultiple edges are not allowed, then the Π-hypothesis is still needed. As a side note, Dorogovtsev et al. [7] also considered the effect of accelerating growth, which is proportional to the power of the tie variable t at each tie step. However, this destroys the scale-free feature and degreedistribution stability of the networ. Afterwards, Bollobás [8] ade a general coent on the BA odel: Fro a atheatical point of view, however, the description above, repeated in any papers, does not ae sense. The first proble is getting started. The second proble is with the preferential attachent rule itself, and arises only for 2. In order to prove results about the BA odel, one ust first decide on the details of the odel itself. It turns out to be convenient to allow ultiple edges and loops. Consequently, he and his coauthors recoended a so-called LCD odel, as follows: We start with the case. Consider a fixed sequence of vertices v, v 2,. We shall inductively define a rando graph process {G t } t 0 so that G t is a directed graph on {v i : i t}, as follows. Start with G 0 the graph with no vertices, or with G the graph with one vertex and one loop. Given G t, for G t by adding the vertex v t together with a single edge directed fro v t to v i, where i is chosen randoly with d t G vs 2t, s t Πi s 2t, s t. For > we define the process {G t } t 0 by running the process {G t } on a sequence v, v 2, ; the graph G t is fored fro Gt by identifying the vertices v, v 2,, v to for v, identifying v +, v +2,, v 2 to for v 2, and so on. For graph G n, let #n d be the nuber of vertices of G n with in-degree equal to d, i.e., with total degree + d, and set a,d 2 + d + d + + d Bollobás et al. [9] rigorously proved the following result: li n E[#n d]/n a,d. Then, based on the artingale theory, they proved that # n d/n converges to a,d in probability. It has been observed that ost real-world and siulated networs follow certain rules to add or reove their vertices and edges, which are not entirely rando. More iportantly, at each tie step, these rules are applied only to the previously fored networ, therefore the process has proinent Marovian properties. Shi et al. [0] established a close relationship between the BA odel and Marov chains. According to the evolution of the BA odel, the degree i t of vertex i at tie t constitutes a nonhoogeneous Marov chain as tie evolves. Thus, all vertices together for a faily of Marov chains. Consequently, based on the Marov chain theory, starting fro an initial distribution and iteratively ultiplying the state-transition probability atrices, the final networ degree distribution can be easily obtained. Lately, Shi et al. [] developed an evolving networ odel by using an anti-preferential attachent echanis, which can generate scale-free networs with power-law exponents varying between 4. There are several odified and generalized BA odels in the literature, including such as the local-world BA odel [2], which will not be listed and reviewed here. All in all, the BA odel indeed is a breathrough discovery with significant ipact on networ science today. Therefore, it is quite iportant to support the odel with a rigorous atheatical foundation. It is clear fro the above discussions that two ey questions need to be carefully answered for the BA odel: For the case of 2, can one find a schee of adding new edges fro the new vertex to the existing ones that has a probability precisely equal to Π? This is the ey of the BA odeling. 2 Does the steady-state degree distribution of the networ exist and, if so, what is it? This is the ey to the validity of the ean-field, rate-equation, aster-equation, and Marov-chain approaches. The present paper will give coplete answers to these two questions. Degree-distribution stability. To start, consider the first question. Recall that Hole and Ki [3] proposed a schee for new edge connection: When a new vertex coes into the networ, the first edge connects to an existing vertex with the preferential attachent probability Π. After that, the rest edges randoly connect with probability p to the vertices in the neighborhood of the vertex that the first edge was connected to, or connect with probability p to those vertices that the first edge did not connect to. Here, consider this approach with p in the following scenario: When a new vertex

3 3 coes into the networ, the first edge connects to an existing vertex with the specified preferential attachent probability Π, sae as above. Yet, the rest edges siultaneously connect to vertices randoly chosen fro inside the neighborhood of the vertex that the first edge was connected to. By rando sapling theory this is equivalent to the above Hole-Ki schee which continually connects the edges to vertices randoly chosen fro inside the neighborhood without allowing ultiple edges. For this special schee, the following result can be rigorously proved. Proposition. For the BA odel with the above special attachent schee, if vertex i has degree i t at tie t, then the probability that vertex i receives a new edge fro the new vertex at tie t + is exactly equal to Π i. Proof. Let P i t+ be the probability of vertex i receiving a new edge fro vertex t + at tie t +. Then, P i t + i t j jt + l O it i t j jt + l O it l t C 2 l t j jt C l t j jt it j jt, where C 2 l t C lt!/[ 2! l t +!] l t l t!/[! l t +!] l t, which is the probability of choosing vertex i, aong the vertices that were randoly chosen fro inside the neighborhood O l t of vertex l, to perfor siultaneous connections. The Proposition answers the first question posted above and shows that the special Hole-Ki preferential attachent schee is one way to ipleent the Πhypothesis. In order to prove the degree-distribution stability of the general BA networ, the BA odel is specified first. Start with a coplete graph with 0 vertices, which has a total degree N 0 0 0, and denote these vertices by 0,,, respectively. In all the following derivations, the Π-hypothesis will be assued. The general BA networs will be further discussed in the last section below. Following Dorogovtsev et al. [6], consider the degree i t as a rando variable, and let P, i, t P { i t } be the probability of vertex i having degree at tie t, and oreover let the networ degree distribution be the average over all its vertices at tie t, naely, P, t P, i, t. t + 0 i 0,i 0 Recall that i t is a rando variable for any fixed t and it is a nonhoogeneous Marov chain for variable t [0]. Under the Π-hypothesis, the state-transition probability of this Marov chain is given by, l 2t+ N 0 P { i t + l i t }, l + 2t+ N 0 0, otherwise, where, 2,, + t i, and i, 2,. The existence of the steady-state degree distribution for this specified BA networ can be proved in three steps as follows. Detailed derivations are supplied in the Appendix of the paper.. Consider the first-passage probability of the Marov chain: f, i, t P { i t, i l, l, 2,, t }. Then, the relationship between the first-passage probability and the vertex degrees is established. Lea. Under the Π-hypothesis, for the BA odel with >, f, i, s P, i, s P, i, t si+ f, i, s, 2 2s + N0 t.3 2. Under the Π-hypothesis, using the state-transition probability of the Marov chain, one first finds the expression of P, t, as follows: P,t t Y i 2 2i + N 0 t 6 4 P, + X t Y l! i + 0 i lq j t + 0 2i + N 0 i " Xt + 0P, + l++ 0 2j+ N 0! ly l j «j+ 0 j ! # Then, one can show the existence of the liit li P, t by using the following classical Stolz- Cesáro Theore in Calculus. Stolz-Cesáro Theore [4]. In sequence { xn y n }, assue that {y n } is a onotone increasing sequence with y n. If the liit li n x n l +, then li n x n+ x n y n+ y n yn l. l exists, where Lea 2. Under the Π-hypothesis, for the BA odel, the liit li P, t exists and is independent of the initial networ: P li P, t > Under the Π-hypothesis, siilarly, one finds the expression of P, t using the first-passage probability of the Marov chain, and then shows the existence of the liit li P, t by using the Stolz-Cesáro Theore, if the liit li P, t exists. Lea 3. Under the Π-hypothesis, for the BA odel with >, if the liit li P, t exists then the liit li P, t also exists: P li P, t P >

4 4 Finally, by atheatical induction, it follows fro Leas 2 and 3 that the steady-state degree distribution of the specified BA networ exists. To this end, by solving the difference equation 5 iteratively, one arrives at the following conclusion. Theore. Under the Π-hypothesis, for the BA odel with, the steady-state degree distribution exists, independent of the initial networ, and is given by P > 0. 6 Clearly, this degree distribution forula is consistent with the forula obtained by Dorogovtsev et al. [6] and Bollobás et al. [9], which allow ultiple edges and loops. Discussion. Bollobás [8] once discussed the BA description the Π-hypothesis of preferential attachent in detail. His result gives a range of odels fitting the BA description with very different properties. When 2, as a new vertex coes in, it is no proble for its first edge to preferentially connect to an existing vertex. But what about the other new edges? This question was not carefully addressed before. Clearly, after the first edge has been connected fro the new vertex to an existing vertex, the preferential attachent probability Π is no longer the sae if later operations do not allow ultiple edges and loops. It is also clear that when 2, the probability of vertex i receiving a new edge is always greater than Π. But what is it? On the other hand, it is also possible that the probability of vertex i receiving a new edge depends on other vertex degrees. Barabási always ephasizes the Π-hypothesis but did not discuss this how question either. Thus, two questions arise: For the BA odel, or for any other BA-lie odel, how to prove the degree-distribution stability if the Π-hypothesis holds only approxiately? 2 Is there a preferential attachent schee for 2 such that the probability of vertex i receiving a new edge is independent of other vertex degrees? To answer these two questions, a new preferential attachent schee is proposed and discussed in [5], where a new vertex will be siultaneously connected to different vertices and it is assued that the preferential attachent probability Π is proportional to the su of the degrees i,, i of those vertices. They showed that the probability that the existing vertex i received an edge fro the new vertex is independent of other vertex degrees, naely, it 0 + t i t t 2t + N t, Π t+ where 0 is the nuber of vertices and N 0 is the total degree in the initial networ. Consequently, under the a t i t + b t + o it,t-hypothesis and soe ild conditions, they proved the degree-distribution stability of Barabási-Albert type networs. Especially, the powerlaw exponent of the networ degree distribution in this new preferential attachent schee is γ 2 +. Finally, it should be ephasized that the theory and schee developed in this paper has great generality [6], in the sense that it can be applied to any BA-lie odified and generalized odels, such as the LCD odel of Bollobás et al. [9], the attraction odel of Dorogovtsev et al. [6], the local-world BA-lie odel of Li and Chen [2], and the evolving networ odel of Shi et al. [], etc. We suarize the results and findings in this paper as follows: Our proving ethod differs fro the one based on artingale theory, and can be applied to any other scale-free types of coplex networs; 2 We do not need to change the BA odel, e.g., to allow ultiple edges and loops; 3 We provide a special Hole- Ki preferential attachent schee such that the Πhypothesis holds. This research was supported by the National Natural Science Foundation under Grant No , and by the NSFC-HKRGC Joint Research Projects under Grant N-CityU07/07. Eail address: Eail address: Eail address: Eail address: [] Watts D. J. and Strogatz S. H., Nature 393, 998, [2] Erdös P. and Rényi A., Publications Matheaticae 6, 959, [3] Barabási A.-L. and Albert R., Science 286, 999, [4] Barabási A.-L., Albert R. and Jeong H., Physica A 272, [5] Krapivsy P. L., Redner S. and Leyvraz F., Phys. Rev. Lett. 85, 2000, [6] Dorogovtsev S. N., Mendes J. F. F. and Sauhin A. N., Phys. Rev. Lett. 85, 2000, [7] Dorogovtsev S. N. and Mendes J. F. F., Phys. Rev. E 63, 200, 0250 [8] Bollobás B., Handboo of Graphs and Networs: Fro the Genoe to the Internet Bornholdt S. and Schuster H. G. eds., Wiley-VCH, 2002, -34 [9] Bollobás B., Riordan O. M., Spencer J. and Tusnády G., Rando Structures and Algoriths 8, 200, [0] Shi D. H., Chen Q. H. and Liu L. M., Phys. Rev. E 7, 2005, [] Shi D. H., Liu L. M., Zhu X. and Zhou H. J., Europhys. Lett. 76, 2006, [2] Li X. and Chen G. R., Physica A 328, 2003, [3] Hole P. and Ki B. J., Phys. Rev. E 65, 2002, [4] Stolz O., Vorlesungen uber allgeiene Arithetic, Teubner, Leipzig 886 [5] Hou Z. T. et al., On the degree-distribution stability of Barabási-Albert type networs, 2008, preprint.

5 arxiv: v [ath.pr] 9 May 2008 Appendix to Degree-distribution stability of scale-free networs Zhenting Hou Xiangxing Kong Dinghua Shi,2 Guanrong Chen 3 School of Matheatics, Central South University, Changsha 40083, China 2 Departent of Matheatics, Shanghai University, Shanghai , China 3 Departent of Electronic Engineering, City University of Hong Kong, Hong Kong, China May 0, 2008 To provide a rigorous proof of the degree-distribution stability of the scalefree networ generated by the BA odel, soe paraeters are specified as follows: i start with a coplete graph with 0 vertices, which has the total degree N 0 0 0, and denote these vertexes by 0,,, respectively; ii assue that at each tie step t, the probability of the new vertex connecting to an existing vertex i is exactly equal to Π i t. Here, in ii, the preferential attachent probability is siultaneously proportional to both the degree i t of the existing vertex i and the nuber of new edges that the new vertex brings in, at tie t. For notational convenience, this assuption will be referred to as the Π-hypothesis below. Observe that the degree i t of vertex i at tie t is a rando variable [6]. Let P, i, t P { i t } denote the probability of vertex i having degree at tie t, and define the degree distribution of the whole networ by the average value of probabilities of vertex degrees P, t t + 0 i 0,i 0 P, i, t. Observe also that the degree i t as a process in tie t is an nonhoogeneous Marov chain [0]. Thus, for, 2,, t + i, the state transition probabilities of the Marov chain, under the Π-hypothesis, are given by, l 2t+ N 0 2t+ N 0 P { i t + l i t }, l + 0, otherwise. 2

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